Combinatorial Game Theory
who wins, and by how much
Concept

Conjunctive compound — where it appears

Several positions played at once with a move made in every one of them, ending the moment any single component ends. Grundy values say nothing about it; what decides it is the smallest remoteness among the parts.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as selective compound — the same set of essays touches all of them, so they are one junction rather than several.

Three ways to add the same games. One list of components, added three different ways. Under the disjunctive rule a move is a move in exactly one part; under the conjunctive rule it is a move in every part at once, and play stops as soon as any part runs out; under the selective rule it is a move in any non-empty set of parts. The outcomes are computed by search from each rule's own definition.

Three ways to add the same games

A move in exactly one component is a choice, not a law. Move in every component at once and the game is different; move in any set of them and it is different again. The same two positions, added three ways, give three different answers — and only one of the three has values that add.

sums · Disjunctive sum Two numbers from the same tree. Four subtraction games, each with its Grundy sequence and its remoteness sequence. The Grundy value decides a disjunctive sum and the remoteness decides a conjunctive one; the only thing they always agree about is which heaps are losses for the player to move.

How long it lasts

Move in every component at once and the game ends the moment any one of them does. Grundy values say nothing about that game; what decides it is the remoteness, a second number computed from the same tree that measures how long a component can be made to last. Over 2,268 positions the rule is right every time, and the two numbers determine each other in neither direction.

sums · Remoteness The number nobody needs. The shortened selective compound — move in any non-empty set of components, and the game stops as soon as any one component stops — solved directly on 1,176 three-heap positions across four subtraction sets, with four predictions beside it. The suspense number was introduced for this compound and it is right; so are three cheaper things, and the shortening leaves the winner unchanged.

The number nobody needs

The compound theory carries a third quantity — the suspense number — computed by the remoteness recursion with both preferences reversed, for the compound that stops as soon as any component stops. It governs that compound correctly. So does remoteness, so does the plain Grundy value, and the shortening does not change the winner on any of 1,176 positions.

sums · Remoteness Four rules, asked of compounds made of two different games. Compounds whose two components come from different subtraction games, solved in full and compared with what each rule predicts. The three rules the compound theory supplies are exact on every position; the shortcut a reader carries instead is not.

A compound of two different games

Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.

sums · Remoteness

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchSelective compoundDisjunctive sumGame lengthGrundy valueImpartialOutcome classParityRemotenessSubtraction gameSuspense numberAdditivity

All concepts